Method of deriving an AC waveform from two phase shifted electrical signals

ABSTRACT

The voltage between two objects is measured utilizing an electro-optic crystal exhibiting birefringence in two axes (slow and fast) mutually orthogonal to an optic axis extending between the two objects. Two collimated light beams polarized at an angle to the slow and fast axes is passed through the crystal parallel to the optic axis with one of the collimated light beams retarded relative to the other by about 1/4 wave. The two beams emerging from the crystal are passed through a polarizer and converted to phase shifted electrical signals by photo diodes in electric circuits which regulate the sources of the light beams to maintain the peak magnitudes of the two electric signals constant and equal. As another feature of the invention, a stairstep output waveform representative of the measured waveform is generated in a digital computer from a bidirectional cumulative count of zero crossings of the two electric signals which is incremented or decremented depending upon which of the two electrical signals is leading. Improved accuracy of the output signal is achieved by adjusting the stairstep waveform by the magnitude of the smaller of the two electrical signals, with the sense of the adjustment determined by the relative polarities of the two electrical signals.

RELATED APPLICATION

This is a division of application Ser. No. 250,289, filed Sep. 28, 1988,now U.S. Pat. No. 4,904,931.

BACKGROUND OF THE INVENTION

1. Field of the Invention

This invention relates to deriving a representation of an ac waveformfrom two electrical signals in quadrature utilizing a digital computer.

2. Background Information

Electro-optical systems for measuring electric voltages are known. Forinstance, devices known as Pockel cells utilize certain crystals whichexhibit birefringence, that is a difference in the index of refractionin two orthogonal planes, in the presence of an electric field. Some ofthese crystals, such as, for example, KDP (potassium dihydrogenphosphate), have a fourfold axis of rotary inversion. Such materialshave the property that in the absence of an electric field the index ofrefraction for light propagating along the fourfold axis is independentof the plane of polarization of the light. However, if an electric fieldis applied parallel to the direction of the light, the index ofrefraction for light polarized in one direction transverse to thefourfold axis, known as the fast axis increases and that in anorthogonal direction, also transverse to the fourfold axis, and known asthe slow axis, decreases by an amount which is proportional to thestrength of the electric field. In such Pockel cell devices, if light ispolarized in a plane which forms an angle to these transverse axes, thecomponent of the polarized light in the direction of the slow axis withthe decreased index of refraction is retarded with respect to the othercomponent. If the crystal is aligned with its fourfold axis extendingbetween the objects between which the voltage is to be measured, and thepolarized light is directed parallel to the fourfold axis, the totalretardation will be proportional to the total voltage differentialbetween the two objects. This retardation is typically measured inwavelengths. The retardation is detected in an analyzer and converted toan electrical signal for producing an output representative of themagnitude of the voltage generating the field. Due to the cylic natureof this electrical signal, the output is only unambiguous for voltagesproducing a retardation which is less than the half wave voltage for thecrystal. In KDP, this half wave voltage is about 11300 volts. This typeof device is therefore not suitable for measuring transmission linevoltages which can be 100,000 volts rms and more.

Other types of crystals used in Pockel cells respond to an electricfield in a direction perpendicular to the direction of propagation oflight through the cell. Such cells only provide an indication of thepotential at the intersection of the beam with the field. Thus, a singlecell cannot integrate the potential over the full space between twoobjects, and therefore these devices do not provide an accuratemeasurement of the voltage between the two objects. Systems using thistype of Pockel cell commonly either, (1) measure the potential at onepoint and assume that the potential at all other points between the twoobjects can be derived from this single measurement, or (2) provide somesort of voltage divider and apply a fixed fraction of the line voltageto the cell in an arrangement which maintains the field within the cellconstant. The problem with the first approach is that except for lowimpedance paths, the field along a path is sensitive to the location ofany conducting or dielectric bodies in the vicinity of the path. Thus,if this type of Pockel cell is mounted on the surface of a conductor andthe field measured, the reading would depend on the size and shape ofthe conductor, on the distance from the conductor to ground, on thelocation and potential of any nearby conductors, on the location of anyinsulating or conducting bodies near the sensor or on the ground beneaththe sensor, and on the presence of any birds, rain droplets or snowbetween the sensor and ground. Thus, only under very ideal circumstanceswould accurate measurements be possible with such a system. The problemwith the second approach is in providing an accurate stable voltagedivider.

Optical voltage measuring systems are desirable because they providegood isolation from the voltage being measured. Through the use of opticfiber cables, it is possible to easily and conveniently provide remoteindicators which are not subject to the electrical disturbances whichremote indicators fed by electrical signals must contend with.

There remains, however, a need for an optical system for accuratelymeasuring very large voltages such as, for example, those present inelectrical transmission systems without the use of a voltage divider.

Subordinate to this need is a need for such an optical system which canintegrate the field over the entire space between the objects, such asin the case of the electrical transmission system between line andground.

SUMMARY OF THE INVENTION

These and other needs are satisfied by the invention which is directedto a method and apparatus for deriving a waveform representative of anoriginal waveform, from two electrical signals in quadrature derivedwith the original waveform. In one sense, the original waveform isreconstructed from selected segments of the two electrical signals withthe segments selected in part as a function of the sequence of zerocrossings of the two electrical signals. At another level, therepresentative waveform can be constructed as a stairstep signal havingdiscrete incremental values which increment or decrement with each zerocrossing depending upon which of the two electrical signals is leading.Reversal of direction of the measured waveform can be detected, forinstance, from two zero crossings in succession by one of the twoelectrical signals.

While such a stairstep waveform may be adequate for many applications,the accuracy of such a signal does not reach the 0.1% desired in themeasurement of transmission line voltages which, for example, can be260,000 volts peak to peak or 93,000 volts rms. Accordingly, theinvention includes interpolating between steps of the stairstep waveformusing the instantaneous value of a selected one of the two electricalsignals. The value of the electrical signal which is smaller inmagnitude is always selected for the interpolation. This results inutilization of portions of the component waveforms where the small angleapproximation, that is where the sine of the angle is approximatelyequal to the angle, is valid, and hence the errors introduced by theinterpolation are small.

In order to eliminate erratic indexing of the zero crossing count forrandom behavior of the electrical signals around zero, a dead band iscentered on the zero axis of the electrical signals. When the value ofthe smaller electrical signal enters this dead band, indexing of thezero crossing count is suspended until the signal emerges from the band.If it exits on the opposite side of the band from which it entered, thezero crossing count is indexed. Whether it is incremented or decrementeddepends upon the direction in which the original waveform is moving,which is manifested by which of the quadrature electrical signals isleading. If the electrical signal of smaller magnitude exits on the sameside of the dead band as it entered, the measured waveform has changeddirection and the zero crossing count is not indexed. Preferably, themagnitude of the electrical signal required to exit the dead band isgreater than that required to enter. This hystereses in the width of thedead band prevents erratic behavior at the boundaries.

While this reconstruction of a waveform from phase shifted electricalsignals is particularly suitable for use in the opto-electrical system,which is also part of the invention, for generating waveformsrepresentative of sinusoidal voltage waveforms of large magnitude, italso has applicability to reconstructing other types of waveforms inother applications.

BRIEF DESCRIPTION OF THE DRAWINGS

A full understanding of the invention can be gained from the followingdescription of the preferred embodiment when read in conjunction withthe accompanying drawings in which:

FIG. 1 is a schematic diagram illustrating the principle of operation ofvoltage measuring systems which form a part of the invention.

FIG. 2 is a schematic diagram of a voltage measuring system inaccordance with the invention.

FIGS. 3a, b and c are waveform diagrams illustrating respectively theline to ground voltage to be measured, the waveforms of the phaseshifted electrical signals generated by the opto-electrical measurementsystem of FIG. 2, and the output waveform reconstructed from the phaseshifted electrical waveforms.

FIG. 4 is a diagram illustrating how the output waveform isreconstructed from the phase shifted electrical waveforms.

FIGS. 5A and 5B are flow charts illustrating the program used by thesystem of FIG. 2 to construct the output waveform from the phase shiftedelectrical waveforms in the manner illustrated in FIG. 4.

FIG. 6 is an isometric view with part broken away of apparatus formeasuring line to ground voltages in a high voltage electric powertransmission system in accordance with the invention.

FIG. 7 is an enlargement of a subassembly of FIG. 6.

FIG. 8 is a vertical section through a component which is part of thesubassembly of FIG. 7.

DESCRIPTION OF THE PREFERRED EMBODIMENTS

As is known, the voltage between two spaced points a and b is defined bythe equation: ##EQU1## where E(x) is the field gradient at x and theintegral is independent of path. Thus, in order to accurately measurethe voltage between spaced points a and b, it is necessary that a sensorphysically extend from a to b, interact with the field at every pointalong its length, and change some property so that some parameter variesin an additive fashion allowing the integral to be evaluated. In themeasurement of transmission line voltages, this requires that one end ofa sensor be electrically connected to the transmission line and theother end be electrically connected to ground. Thus, the sensor must beof sufficient length to withstand normal line voltages and any surgeswhich might be encountered.

The present invention utilizes an electro-optic crystal to measure theintegral of the field gradient from point a to b and thus provides atrue value for the voltage between a and b. As mentioned previously,certain crystaline materials having a fourfold axis of rotary inversion,such as KDP (potassium dihydrogen phosphate), have the property that inthe absence of an electrical field, the index of refraction for lightpropagating along the fourfold axis is independent of the direction ofpolarization of the light. However, if an electric field is appliedparallel to the direction of propagation of the light, the index ofrefraction for light polarized in a given direction perpendicular to thefourfold axis increases while the index of refraction of light polarizedin a perpendicular direction decreases by an amount which isproportional to the field. In KDP, the direction parallel to thefourfold axis, which is also called the optic axis, is commonlydesignated as the Z direction, and the orientations of the polarizationfor which the maximum changes in refractive index with electric fieldare observed are commonly designated as the X' and Y' directions.

To understand the principle of operation of such an opto-electricalsensor, reference is made to FIG. 1. In the conventional Pockel celldevice 1, a KDP crystal 3 is aligned with its fourfold axis of rotaryinversion, Z, parallel to the field gradient, F_(g) to be measured. Asingle beam of unpolarized light is incident on a first linear polarizer7. The crystal 3 and first polarizer 7 are arranged such that collimatedpolarized light, 5_(p) exiting the polarizer is propagating parallel tothe Z axis of the crystal and the plane of polarization of the light isat an angle of 45 degrees to the X' and Y' axes of the crystal.

The incident polarized beam 5_(p) can be decomposed into two componentsof equal intensity, one polarized parallel to the X' axis and the otherpolarized parallel to the Y' axis. In the absence of an electric field,these two components will propagate with equal velocities and exit thecrystal 3 in phase with one another. When an electric field is appliedalong the Z axis of the crystal, the refractive indexes, and, as aresult, the velocities of the two components will not be equal, andthere will be a phase shift or a retardation between the two componentswhen they exit the crystal. Since the retardation in any small elementalong the crystal is proportional to the electric field acting on thatelement multiplied by the length of the element, and the totalretardation is equal to the sum of the retardations in all of theelements along the crystal, retardation of the components exiting thecrystal is proportional to Edl, and thus the difference in voltagebetween the two ends of the crystal.

The retardation is usually expressed in wavelengths, that is aretardation of one means the optical path in the crystal is onewavelength longer for one of the components of the beam 5_(p) than forthe other, and is given by the equation: ##EQU2## where r₆₃ is anelectro-optic coefficient, n_(z), is the refractive index for lightpropagating along the Z axis, λ is the wavelength of the light invacuum, and V is the difference in voltage in the two ends of the of thecrystal. While these parameters are known and the retardation can becalculated, it is usually more convenient to combine them in a singleparameter, the half-wave voltage, V_(h), defined by the equation:##EQU3## V_(h) is usually determined as part of the calibration of thesensor. If the two components of the beam 5_(p) exiting the crystal arepassed through a second polarizer 9 oriented parallel to the first, theintensity of the beam I exiting the polarizer 9 is related to theretardation by equation:

    I=Io Cos.sup.2 (πΓ)                               (Eq. 5)

where I_(O) is the intensity of the exiting beam with zero retardation:That is, with no voltage difference between the ends of the crystal. Ifthe second polarizer 9 is rotated 90 degrees, then I is given byequation 5 in which the square of the sine function is substituted forthe square of the cosine function.

It is common in such Pockel cell devices described to this point toinsert a fractional wave plate 11 between the crystal and the secondpolarizer 9 to shift the retardation to a linear point on the sine orcosine squared function.

Because of the periodic properties of the sine and cosine functions, adevice as discussed to this point would only provide unambiguous resultsfor voltages less than v_(h). For KDP, at a wavelength of 800 nm, V_(h)is roughly 11,300 volts, and thus such a device cannot be used tomeasure transmission line voltages which are typically around 100,000volts rms line to ground or more.

In order to resolve the ambiguities inherent in the conventional Pockelcell arrangement, and allow measurements at transmission line voltages,the present invention utilizes a second light beam 13 parallel to thebeam 5. This second light beam 13 is polarized by the first polarizer 7to form a second polarized light beam 13_(p) which is passed through thecrystal 3 parallel to the Z axis. This second polarized light beam13_(p) can also be resolved into two components, one parallel to the Zaxis and the other parallel to the Y axis. The second beam exiting thecrystal 3 is also passed through the second polarizer 9 so that theintensity of the second beam exiting polarizer 9 is also related to theretardation by equation 5 if the second polarizer is oriented parallelto the first polarizer 7 or by the sine squared function if the secondpolarizer is orthogonal to the first polarizer. The second light beam 13exiting the crystal 3 is also retarded by a fractional wave plate 15before passing through the second polarizer 9. The fractional waveplates 11 and 15 are selected so that one beam is retarded with respectto the other. In the preferred form of the invention, the one beam isretarded 1/4 wave with respect to the other so that the beams exitingthe second polarizer are in quadrature. This retardation may beaccomplished by utilizing one-eighth wave plates for the fractional waveplates 11 and 13 with their axes 17 and 19 respectively oriented 90degrees with respect to one another. Other arrangements can be used toretard the one light beam 1/4 wave with respect to the other. Forinstance, one beam could be passed through a quarter wave plate whilethe other passes directly from the crystal to the second polarizer.Retarding one beam exactly 1/4 wave with respect to the other simplifiesthe calculation required, but as long as the retardation is about 1/4wave, meaning within about plus or minus 20% of 1/4 wave, satisfactoryresults can be achieved. In addition to 1/4 wave retardation, oddmultiples of 1/4 wave can also be used i.e., 3/4, 5/4 et cetera.

With the two one-eighth wave plates oriented as indicated in FIG. 2, theintensities of the two beams exiting the second polarizer can bedetermined as follows:

    I.sub.1 =Io Cos.sup.2 (πΓ+π/8)                 (Eq. 6)

and

    I.sub.2 =Io Cos.sup.2 (πΓ-π/8)                 (Eq. 7)

These two signals are in quadrature and, with the exception of aconstant, allow the unambiguous determination of the voltage applied tothe crystal.

FIG. 2 illustrates schematically a complete voltage measuring system inaccordance with the invention. This system 21 includes the sensor 1comprising the crystal 3, the first and second polarizers 7 and 9respectively, and the one-eighth wave plates 11 and 15. The system 21also includes first and second light sources 23 and 25 which generatethe two collimated light beams 5 and 13 respectively. The light source23 includes a light emitting diode (LED) 27. Light produced by the LED27 is transmitted by optic fiber cable 29 and passed through collimatinglens 31 to produce the first collimated light beam 5. Similarly the LED33 in second light source 25 produces light which is transmitted by theoptic fiber cable 35 and passed through collimating lens 37 to producethe second collimated light beam 13. Light from the first beam 5 exitingthe second polarizer 9 is gathered by lens 39 and conducted throughfiber optic cable 41 to a first electronic circuit 43. Similarly, thesecond beam exiting the second polarizer 9 is focused by lens 45 onfiber optic cable 47 which directs the light to a second electroniccircuit 43.

The electronic circuits 43 are identical and include a photo diode 49which converts the light beam carried by the optic fiber cables 41 or 47respectively into an electrical current. The electronic circuits 43include a transimpedance amplifier 51 which provides a low impedanceinput to a peak detector 53. Peak detector 53 includes a diode 55 whichfeeds a capacitor 57 shunted by leak resistor 59. The peak detector alsoincludes a buffer amplifier 61 to prevent the peak detector from beingloaded by the following stage. The following stage 63 acts as a summingamplifier, integrator, and a driver for the respective LED 27 or 33. Itincludes a pair of resistors 65, and an operational amplifier 67 shuntedby an integrating capacitor 69. An output circuit includes a pair ofresistors 71 and 71' (equal to twice 71 in value), and an outputamplifier 73. A reference voltage -e_(r) is applied to the summingcircuits formed by the resistors 65 and 71-71'.

The electronic circuits 43 operate as follows: Light exiting the secondpolarizer 9 and transmitted via the optic fiber cable 41 or 47respectively is converted to an electrical current signal by the photodiode 49. The peak detector 53 generates a signal which represents thepeak value of this electrical current. The peak value signal is comparedwith the reference signal through the resistors 65 connected to theinverting input of the operational amplifier 67. Since the diode 55assures that the peak signal is positive, and since the reference signal-e_(r) is negative, these two signals are compared and the error betweenthe two is applied to the integrator formed by the operational amplifier67 and the capacitor 69. This integrated error signal is used to drivethe LED 27 or 33 respectively of the light sources for the first andsecond light beams. Thus, the circuits 43 are feedback circuits whichregulate the intensity of the respective light beam so that the peakvalues of the current signals generated by these light beams through thephoto detectors 49 remain constant and equal to the reference voltageand, hence, equal to each other. The summing amplifier 73 and voltagedividing resistors 71 subtract the reference voltage from theunidirectional currents produced by the photodetectors 49 to producebipolar voltage output signals e₁ and e₂ respectively in response to thefield applied to the crystal 3. The analog signals e₁ and e₂ areperiodically sampled by an analog to digital converter 75 for input intoa digital computer 77. The digital computer 77 reconstructs the voltagewaveform from the two signals e₁ and e₂ for presentation on an outputdevice 79. The output device 79 can be, for instance, a digital readout,and/or can be a recorder which generates a permanent log of the measuredvoltage waveform.

Waveforms a, b and c of FIG. 3 illustrate on a comparative time basisthe voltage waveform V_(I) to be measured, the quadrature electricalsignals e₁ and e₂ generated in response to the voltage waveform a by thesystem of FIG. 2, and the output waveform V₀ generated by the system ofFIG. 2 which is representative of the voltage waveform a.

FIG. 4 illustrates the manner in which the waveform c in FIG. 3 isconstructed from the quadrature electrical signals e₁ and e₂ forming thewaveform b in FIG. 3. Essentially the method comprises maintaining abidirectional count of the number of zero crossings of the twoelectrical signals e₁ and e₂. In the example given in FIG. 4, the count,n of such zero crossings is shown across the top of the figure. Thecount n is incremented as the voltage waveform represented by e₁ and e₂is becoming more positive (or less negative) and is decremented as themagnitude of the incremented waveform is becoming more negative (or lesspositive). The direction in which the waveform is moving is determinedby which of the quadrature signals is leading. Reversal of the directionof the voltage waveform results in a switch in which signal e₁ or e₂ isleading and can be detected by two successive zero crossings by the samesignal e₁ or e₂.

A stairstep approximation of the voltage waveform indicated by the trace81 in FIG. 4 can be generated from the cumulative count n of the zerocrossings. In the particular sample shown, the output stairstep waveformis generated as a function of twice the accumulated count n as shown bythe scale on the left side of the trace 81 in FIG. 4.

This stairstep approximation 81 of the original voltage waveformgenerating the field applied to the optoelectrical sensor can beadequate for many purposes. However, where more accurate reproduction ofthe original voltage waveform is required, such as in monitoring thevoltage of high power transmission lines where an accuracy of 0.1percent is required, interpolation must be made between the stairstepvalues of the output signals generated by the accumulated zero crossingcount n. This smoothing of the output waveform is achieved by adding orsubtracting the instantaneous value of a selected one of the quadraturesignals e₁ and e₂ to the stairstep value. The selected signal is the oneof the two signals, e₁ and e₂, which is smaller in magnitude at thegiven instant. Thus, the magnitude of the signal e₁ or e₂ which isbetween the traces 83 and 85 in FIG. 4 is selected. This results inutilization of portions of the waveforms e₁ and e₂, where the smallangle approximation, that is, where the sine of the angle isapproximately equal to the angle, is valid. As can be seen in FIG. 4,the technique essentially results in the stringing together of thesegments of the quadrature signals e₁ and e₂ to reconstruct the originalvoltage waveform.

In order to avoid random indexing of the bidirectional cumulative countn of zero crossings which could occur with small signals, a band iscreated around the zero axis as indicated by the lines 89 and 91 in FIG.4. Zero crossings are not counted while a signal e₁ or e₂ is in thisdead band. Instead, a determination is made when the signal again leavesthe dead band as to whether the zero crossing n should be indexed. Ifthe quadrature signal exits the dead band on the same side that itentered, then the target signal has changed direction and n should notbe indexed. If the quadrature signal exits the dead band on the oppositeside from which it entered, then there has been a zero crossing and n isindexed. While a quadrature signal is within the dead band, itsmagnitude is continued to be used to interpolate between the stairsteps.If the quadrature signal crosses the zero axis, the sign of theincrement which is added or subtracted to the stairstep is changed toreflect this transition. This delay in the indexing of the zero crossingcount n until the quadrature signal leaves the dead band results in aslight shift in time of the stairstep signal, as indicated at 93 in FIG.4. The width of the dead band should be set as wide as possible withouthaving the instantaneous values of both e₁ and e₂ fall within the bandat anytime. Preferably, the boundaries of the dead band are expanded to89'-91' once a signal is within the band. This introduces hysteresisinto the dead band which aids in assuring a positive transition into andout of the dead band.

FIGS. 5a and 5b constitute a flow chart of the program employed by thedigital computer 77 to reconstruct the voltage waveform sensed by thesensor from the quadrature electrical signals e₁ and e₂, in the mannerdiscussed in connection in FIG. 4. As discussed previously, the analogquadrature signals e₁ and e₂ are applied to an analog to digitalconverter to generate digital samples of the instantaneous value ofthese waveforms for processing by the digital computer. The samplingrate should be sufficiently rapid that, at the maximum slew rate of thevoltage signal being measured, which usually occurs at the its zerocrossing, at least one data sample for e₁ or e₂ falls within he deadband. The sampling rate for the 60 hz voltage signal was 400 KHz in theexemplary system. As indicated at block 101 in FIG. 5a, the programwaits for each new data sample. Two variables, E₁ and E₂, are set equalto the current instantaneous value of the quadrature signals e₁ ande.sub. 2 respectively at 103 when each new data sample is acquired.Another variable S which indicates whether the signs of the currentvalues of E₁ and E₂ are the same is set, and that indication is saved asan additional variable SS in block 105.

A flag Q₁, which has a value of 1 if the preceding E₁ was inside thedead band defined by the lines 89-91 (entering) and 89'-91' (exiting) inFIG. 4, is checked block 107. If the preceding E₁ was inside the bandand the present value of the first quadrature signal remains inside theband as determined in block 109 (using the larger boundary of lines89'-91' equal to 0.24 ER), then the variable S, which will be recalledis an indication of whether the signs of E₁ and E₂ are the same, is setat 111 equal to S0 which is the value of S for the preceding data point.

If E₁ has emerged from the dead band, then the flag Q₁ is made equal tozero at 113 and it becomes necessary to determine whether the zerocrossing count, n, should be indexed, and if so, in which direction.This is accomplished by setting another variable A at 115 to indicatewhether the sign of the present E₁ is equal to the sign of E₁₀₀ which isthe last value of E₁ before E₁ entered the dead band. For the purpose ofthis determination, the sign of E₁ is +1 if E₁ has a positive value, or-1 if it has a negative value so that A can have a value of -2, +2 orzero. If the signs of E₁ and E₁₀₀ are the same as determined in block117, E₁ has emerged from the same side of the dead band at which itentered and hence there has been no zero crossing. If these signs arenot the same, then E₁ has emerged from the opposite side of the deadband from which it entered and hence there was a zero crossing and nmust be indexed. If the sign of A is the same as the sign of E₂ asdetermined in block 119 then the voltage is going up and a variable D isset equal to one at 121. If these signs are not equal, then the voltageis going down and D is set at equal to minus one at 123. The cumulativezero crossing count n is then indexed in the proper direction at block125.

If it was determined back at block 107 that the preceding instantaneousvalue of E₁ was outside the dead band, then a check is made at 127 todetermine if the present value of E₁ is within the dead band (using thenarrower boundary defined by lines 89-91 in FIG. 4). If E₁ is now in thedead band, then the flag Q₁ is made equal to one, E₁₀₀ which is avariable equal to the last value of E₁ before the band was entered ismade equal to E₁₀ which is the preceding value of E₁, and S which, itwill be recalled, is an indication of whether the signs of E₁ and E₂ arethe same is made equal to S0 which is the value of S for the last point,all as indicated at block 129. If E₁ remains outside of the dead band,then a determination is made at block 131 as to whether the precedingvalue of E₂ was outside the band. If it was, and the present value of E₂is within the band as determined at block 133 (using the enteringboundary lines 89-91 in FIG. 4), then a flag Q₂ is made equal to one,the last value of E₂ before it went into the band is saved, and S ismade equal to S0 all in block 135. If it was determined in block 133that E₂ was not within the band, then both E₁ and E₂ remain outside thedead band and the program proceeds to the calculation of the presentvalue of the voltage signal in the manner discussed below.

If it was determined at 131 that E₂ was inside the band at the previousdata point, a determination is made at block 137 whether it is stillwithin the band. If it is, S is set equal to S0 in 139. If E₂ has nowemerged from the band, then the flag Q₂ is set equal to zero at 141. Adetermination is then made in blocks 143 and 145 using the variable A ina manner similar to that described in connection with blocks 115 and117, to determine whether the zero crossing count n should be indexed.If E₂ has emerged from the opposite side of the dead band from that fromwhich it entered, then n is incremented or decremented in box 147, 149,151 and 153 using the same technique as described in connection withemergence of E₁ from the dead band. That is, n is incremented ifwaveform e₁ leads e₂ and hence the voltage being measured is increasing,or n is decremented when the measured voltage is decreasing.

Turning to FIG. 5b, a determination is made in block 155 whether therehas been a zero crossing by determining if SS, which is the saved sign,is equal to S, which was set equal to S for the preceding data point ifeither E₁ or E₂ is currently in the band. If there has been a zerocrossing, a variable P is set equal to minus one at 157, otherwise P isset equal to plus one at 159.

If E₁ and E₂ are not of the same sign as determined in block 161 and E₂is of smaller magnitude as determined in block 163, then a voltage EC iscalculated using the cumulative zero crossing count n and the currentmagnitude E₂ in the equation in block 165. However, if E₁ and E₂ are notof the same sign but E₁ is smaller than E₂, then E₁ is used with n tocalculate EC using the formula in block 167. When E₁ and E₂ are of thesame sign, as determined at 161 and E₁ is smaller, as determined at 169,then E₁ is used with n to calculate the value of EC in block 171. On theother hand, if E₂ is the smaller of the two signals which are not of thesame sign, then E₂ is used with n to calculate EC in block 173. As willbe noticed, the first term in the equations for EC in blocks 165 through171 determines the stairstep value from the cumulative count of zerocrossings n, and the second term provides the interpolation based uponthe magnitude of the selected quadrature signal.

The calculated voltage EC is then multiplied by a scaling factor inblock 173 to determine the instantaneous magnitude, E, of the measuredvoltage.

It is convenient to chose e_(r), the reference voltage used in theelectronic circuits, equal to 2.828 volts, so that that the quantity0.3535×e_(r) =1 and 2×0.3535/e_(r) =1/4 and hence the computations inblocks 165, 167, 171, 173 and 175 are simplified.

The measured voltage is unambiguously determined by this procedureexcept for a constant error. This error is the result of the uncertaintyof the initial value of n when the program is started. It is noted thatn is an integer but otherwise arbitrary. If n can be set equal zero whenthe voltage is zero, then subsequent voltage measurements will becorrect. In general this is not possible and one must adjust n ininteger steps until the average value of the calculated voltage over onecycle is zero. After n is properly adjusted, the calculated voltageswill be correct until the program is interrupted.

The program is completed by storing the present values of S, E₁ and E₂as the last value in block 177 in preparation for the next computationof E. The program then loops back to the beginning and waits for thenext input of data.

FIGS. 6 through 8 illustrate a practical embodiment of a sensor 1,mounted in an insulation column 201 which is cut away to show themounting of the sensor. An upper supporting tube 203 is connected to atransmission line (not shown) and a lower supporting tube 205 isconnected to ground. Both tubes are electrically conducting and providecontact between the ends to the sensor 1 and the line and groundrespectively through mounting discs 207 constructed from electricallyconducting transparent material such as NESA glass. Crystal 209 and thepolarizers 211 and 213 are made with a circular cross section ratherthan square as in FIG. 1 and 2 to reduce the electrical stresses.

As shown more clearly in FIG. 7 for the second polarizer 213, twocylindrical collimators 215 are mounted on one flat end face of thecylindrical polarizer and rectangular one-eighth wave plates 217 and 219are mounted against the opposite end. The collimators 215, which focusthe light beams received from the second polarizer 213 on the opticfiber cables 221, are shown broken away in FIG. 8. Each collimator 215is formed from two pieces 223 and 225 of low refractive index glass,such as fused silica and one piece of high index glass 227 such as SF59.The radius of the curved surface of 227, the thickness of 227 and thelength of 225 are chosen so that a bundle of parallel light entering 223is focused on to the end of optic fiber 221 and the rays from the edgeof the bundle strike the fiber. More particularly, these parameters achosen so that the radius of the bundle of light divided by the focallength of the lens is equal to or greater than the numerical aperture ofthe fiber divided by the refractive index of the lower refractive indexglass. The collimators at the other end of the sensor are similarlydesigned, but operate in the reverse direction to transform lightreceived from the fiber optic cable into the bundle of parallel lightwhich is passed through the first polarizer 211. This form of acollimator is necessary since in order to withstand the high electricalstresses during operation, and especially impulse tests, the insulator201 is filled with oil or pressurized sulfur hexafluroide (SF6), andthus the optical system cannot have any glass air interfaces.

While specific embodiments of the invention have been described indetail, it will be appreciated by those skilled in the art that variousmodifications and alternatives to those details could be developed inlight of the overall teachings of the disclosure. Accordingly, theparticular arrangements disclosed are meant to be illustrative only andnot limiting as to the scope of the invention which is to be given thefull breadth of the appended claims and any and all equivalents thereof.

I claim:
 1. A method of constructing a waveform signal representative ofan original waveform which has been resolved into first and secondbipolar electrical waveform signals substantially in quadrature andhaving substantially constant and equal peak magnitudes comprising thesteps of:periodically generating data samples representative of theinstantaneous value of said first and second electrical signals;determining from said data samples zero crossings of said first andsecond electrical signals, accumulating a bidirectional cumulative countof said zero crossings, with the direction of counting being determinedby which of said first and second electrical signals is leading; andgenerating an output signal representative of the original waveform fromsaid cumulative count of zero crossings.
 2. The method of claim 1including:freezing said cumulative count of zero crossings when themagnitude of the current data sample of either of said electricalsignals falls within a band representative of a magnitude less than apreselected value; and unfreezing said cumulative count of zerocrossings in response to the magnitude of the current data sample ofeither of said electrical signals exiting said band and indexing thecumulative count in response thereto, but only when the electricalsignal which exits the band, exits with a polarity which is opposite thepolarity of that electrical signal when it entered the band.
 3. Themethod of claim 2 wherein the rate at which said data samples aregenerated is such that the data sample from only one electrical signalat a time can be within said band.
 4. The method of claim 2 wherein saidband is representative of a magnitude less than a preselected firstvalue when freezing said cumulative count and is representative of amagnitude less than a preselected second value which is greater thansaid preselected first value when unfreezing said cumulative count. 5.The method of claim 3 including adjusting the magnitude of the outputsignal generated from said cumulative count of zero crossings by themagnitude of the current data sample of a selected one of saidelectrical signals.
 6. The method of claim 5 wherein the magnitude ofthe output signal generated from said cumulative count of zero crossingsis adjusted by the magnitude of the current data sample of theelectrical signal which is currently smaller in magnitude.
 7. The methodof claim 6 wherein the magnitude of the output signal generated fromsaid cumulative count of zero crossings is adjusted by the magnitude ofthe data sample of the electrical signal which is smaller in magnitudeapplied in a sense which is determined by the relative polarities of thecurrent data samples of the electrical signals.
 8. The method of claim 7including:freezing the cumulative count of zero crossings when themagnitude of the current data sample of either of said electricalsignals falls with a band representative of a magnitude less than apreselected value; and reversing the sense of adjusting the magnitude ofthe output signal by the magnitude of the smaller electrical signal whenthe smaller electrical signal makes a zero crossing while saidcumulative count is frozen, and unfreezing said cumulative count inresponse to the magnitude of the current data sample of either of saidelectrical signal exiting the band, and indexing the cumulative count inresponse thereto, but only when the electrical signal which exits theband, exits the band from the side opposite to that at which it entered.